Question

Problem 6. Find the reminder of 3282 after divided by 12.
Hint: You can try doing it directly, or otherwise use that 12 = 3: 4 noting
that gcd(3; 4) = 1 and apply the results I posted at URCourses under the
`Assignment 7' section.

Helpful results:

Proposition 1: Let m1;m2 2 Z with m1;m2 > 0 with gcd(m1;m2) = 1. Let
a 2 Z. Then
[ [ m1ja ^ m2ja ] () m1m2ja ]
Equivalently, the above can be expressed in the following way:
[ [ a 0 (mod m1) ^ a 0 (mod m2) ] () a 0 (mod m1m2) ]
Example of application of proposition 1:
[ [ 4 divides a ^ 3 divides a ] () 12 divides a ]
Proposition 2: (Generalization of Proposition 1.) Let m1;m2; : : : ;mk 2 Z
with mi > 0 for i = 1; ; k, with the mi's beeing mutually coprime (that is,
with gcd(mi;mj) = 1 for every i 6= j). If a; b 2 Z, and m = m1 mk then:
m1ja ? b
m2ja ? b
...
mkja ? b
9>>>>=
>>>>;
() mja ? b
Equivalently, the above can be expressed in the following way:
a b (mod m1)
a b (mod m2)
...
a b (mod mk)
9>>>>=
>>>>;
() a b
?
mod m

Example of application of proposition 2:
[ [ 2 divides a ^ 3 divides a ^ 7 divides a ] () 42 divides a ]

Answer 1

So 3282 mod 12 I Ž 2.3.547 mod 12 E 2.3.7. meedia at .: 547 med 12 Ē med 12 E 2.21 meedia - 42 meel 12 i 6 media =) 6 will be reusendes hehen 3282 divided by 12. I